Modular 9 arithmetic is the arithmetic of the remainders after division by 9.
For example, the remainder for 12 after division by 9 is 3. This is expressed as

12 = 3 (mod 9)

Likewise 25=7 (mod 9) and 9=0 (mod 9). Note that 12+25=37 and that 37=1 (mod 9).
But 3+7=10=1 (mod 9) so the equivalent of the sum of two numbers modulo 9 is
equal to the modulo 9 equivalent of the sum of their modulo 9 equivalents.

The modulo 9 equivalent of 12 is 3 which is also the digit sum of 12. This is
no coincidence. There is a very close relationship between the modulo 9 equivalents
of numbers and their digit sums.

Integer

Digit Sum

Remainder
Function Mod 9

1

1

1

2

2

2

3

3

3

4

4

4

5

5

5

6

6

6

7

7

7

8

8

8

9

9

0

10

1

1

11

2

2

12

3

3

13

4

4

14

5

5

15

6

6

16

7

7

17

8

8

18

9

0

The only difference between the values in the second and third columns is that for
a multiple of 9 the remainder function is zero but in the second column the value is 9.
This is reasonable in that 9=0 (mod 9). Let the remainder function for integer k be expressed as
k%9 and
the second column function as k#9. Rather than make a special case for the multiples of 9
it is more elegant to express the correspondence between the two functions as

k#9 = (k-1)%9 + 1
and
k%9 = (k+1)#9 - 1.

The # function can be generalized to

k#n = (k-1)%n + 1
and hence
k%n = (k+1)#n - 1.

The digit sum function is the special case of the k#n function when n=9.

DigitSum(k) = (k-1)%9 +1

The multiplication table for modulo 9 arithmetic is:

Multiplication Table for Modulo 9 Arithmetic

0

0

0

0

0

0

0

0

0

<

0

1

2

3

4

5

6

7

8

0

2

4

6

8

1

3

5

7

0

3

6

0

3

6

0

3

6

0

4

8

3

7

2

6

1

5

0

5

1

6

2

7

3

8

4

0

6

3

0

6

3

0

6

3

0

7

5

3

1

2

6

4

2

0

8

7

6

5

4

3

2

1

If the 0's were replaced by 9's and the table rearranged so the first column becomes the last column and the first row
becomes the last row the result would be
identical to the table for the sequences of digit sums.

The Rearrangement of the
Multiplication Table for Modulo 9 ArithmeticWith 9 Substituted for 0

1

2

3

4

5

6

7

8

9

2

4

6

8

1

3

5

7

9

3

6

9

3

6

9

3

6

9

4

8

3

7

2

6

1

5

9

5

1

6

2

7

3

8

4

9

6

3

9

6

3

9

6

3

9

7

5

3

1

2

6

4

2

9

8

7

6

5

4

3

2

1

9

9

9

9

9

9

9

9

9

9

This table is identical with the table of digit sums for multiples of numbers. The
conclusion is that digit sum arithmetic is the virtually the same as modular 9
arithmetic except there is a replacement of 0's with 9's. The equivalence of 9 and 0 takes care of a small
problem. The digit sum of all multiples of 9 is 9 except for the case of 0 times 9 which has a digit sum
of 0. So the digit sum of all multiples of 9 is equivalent to 9.

In mathematical terminology there
is an isomorphism between digit sum arithmetic and modular 9 arithmetic. All of
the properties of associativity, distributivity, communtivity, identities and
additive inverses carry over from modular 9 arithmentic to digit sum arithmetic. Multiplicative
inverses exist for some elements but not all so neither entity is a mathematical field.
In the table below the cases of products that yield the multiplicative
identity; i.e., 1; are shown in red.

Multiplication Table for Modulo 9 Arithmetic

0

1

2

3

4

5

6

7

8

0

0

0

0

0

0

0

0

0

0

1

0

1

2

3

4

5

6

7

8

2

0

2

4

6

8

1

3

5

7

3

0

3

6

0

3

6

0

3

6

4

0

4

8

3

7

2

6

1

5

5

0

5

1

6

2

7

3

8

4

6

0

6

3

0

6

3

0

6

3

7

0

7

5

3

1

2

6

4

2

8

0

8

7

6

5

4

3

2

1

As can be seen from the above table inverses exist for 1, 2, 4, 5, 7 and 8 but not
for 0, 3 and 6.